Subdiffusion in a time-dependent force field

Shkilev, V. P.

doi:10.1134/s1063776112030089

Cited by 12 publications

(6 citation statements)

References 25 publications

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“…The fractional Fokker-Planck equation with both time-and space-dependent forces was derived in [7]. Other fractional Fokker-Planck equations have been derived outside of a CTRW framework [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Continuous Time Random Walks with Reactions Forcing and Trapping

Angstmann

Donnelly

Henry

2013

Math. Model. Nat. Phenom.

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One of the central results in Einstein's theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation. Continuous Time Random Walks with Reactions and Forcingwas derived in [12][13][14]. This was extended to non-linear reactions in [15]. An alternative derivation with pure death processes was given earlier in [16].The CTRW model for sub-diffusion exhibits non-ergodic behaviour. The ensemble average mean square displacement scales as a sub-linear power law in time whereas time average mean square displacements scale approximately linearly with time, with different diffusion coefficients for different realisations [17,18]. An alternate model for sub-diffusion is fractional Brownian motion (fBm). This model has a Gaussian probability density function with a time dependent diffusion coefficient and it exhibits ergodic behaviour [19]. In this article we have confined our attention to the CTRW model, however tests to distinguish between different models for sub-diffusive behaviour are an active area of research [18,20].Among the experimental systems that exhibit diffusion, reactions and forcing, the motility of populations of cells is especially interesting because of the confounding aspect of chemotaxis, where particles experience a force in proportion to a chemical gradient. The governing equations to model sub-diffusion with linear reactions and chemotaxis have been derived in [21]. Sub-diffusion with non-linear reactions and chemotaxis was considered in [22].Fractional Fokker-Planck equations and fractional reaction-diffusion equations have become increasingly important because of the growing recognition that anomalous diffusion is ubiquitous [23]. Anomalous sub-diffusion is widespread in biological systems due to traps and crowding effects. In such systems the traps may be structural traps [24, 30] or binding spots [26]. Anomalous sub-diffusion has been measured in protein movements in cells [27], in movement of lipid granules in yeast cells [28] and movements of molecules in spiny neuronal dendrites [29, 30]. The consideration of the possible anomalous diffusion of ions in nerve cells prompted the development of a fractional cable equation to model the electrical...

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Section: Introductionmentioning

confidence: 99%

Continuous Time Random Walks with Reactions Forcing and Trapping

Angstmann

Donnelly

Henry

2013

Math. Model. Nat. Phenom.

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“…Diffusion does not always occur in completely homogeneous conditions, especially in complex systems such as comb structures. Therefore, diffusion equation with a time-dependent diffusion coefficient is more appropriate in these cases due to the geometrical characteristics of the medium [47][48][49][50]. In addition to that, the diffusion problem with a time-dependent diffusion coefficient is a good representative method to track the behavior of particles over time, while the diffusion coefficient can be changed due to many factors, such as the presence of obstacles, interactions between particles, and changing conditions in the medium.…”

Section: Numerical Modelmentioning

confidence: 99%

Tempered fractional diffusion in comb-like structures with numerical investigation

Mokhtar Hefny,

Tawfik

2023

Phys. Scr.

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This paper presents two models for describing anomalous transport in comb-like structures. First, we analytically solve the tempered fractional diffusion model using the Laplace-Fourier technique. The probability distributions along the backbone (x-axis) and branches (y-axis) are represented by the M-Wright and Fox’s H functions. The probability distributions are illustrated according to the time-fractional order α and the tempered parameter λ. Additionally, we determine the mean square displacement to classify the degree of diffusivity in the comb structure based on the values of the time-fractional and tempered orders. Second, we introduce a power-law time-dependent diffusion coefficient as an extension of the comb-like models. In this section, we investigate the solution of the proposed comb-like via numerical simulation and explore the connection between the presence of a time-dependent diffusion coefficient and anomalous transport based on the particle density and mean square displacements.

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“…To obtain more general models, we can use the Markov representation of the random barriers model at the mesoscopic level instead of the non-Markovian equation for the propagator Eq. 15 (13,14). In this approach, it is assumed that at each position in space the particle can be in M different transport states and that the probabilities of being in these states satisfy equations…”

Section: Generalization Of the Multiple Trapping Modelmentioning

confidence: 99%

Kinetic model for fluorescence microscopy experiments in disordered media with binding sites and obstacles

Shkilev

2018

Phys. Rev. E

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A model is proposed that describes the diffusion of molecules in a disordered medium with binding sites (traps) and obstacles (barriers). The equations of the model are obtained using the subordination method. As the parent process, random walks on a disordered lattice are taken, described by the random barriers model. As the leading process, the renewal process that corresponds to the multiple-trapping model is taken. Theoretical expressions are derived for the curves obtained in experiments using fluorescence microscopy (FRAP, FCS and SPT). Generalizations of the model are proposed, allowing to take into account correlations in the mutual arrangement of traps and barriers. The model can be used to find parameters characterizing the diffusion and binding properties of biomolecules in living cells.

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Subdiffusion in a time-dependent force field

Cited by 12 publications

References 25 publications

Continuous Time Random Walks with Reactions Forcing and Trapping

Continuous Time Random Walks with Reactions Forcing and Trapping

Tempered fractional diffusion in comb-like structures with numerical investigation

Kinetic model for fluorescence microscopy experiments in disordered media with binding sites and obstacles

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